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Research article

Simulating chi-square data through algorithms in the presence of uncertainty

  • Received: 11 February 2024 Revised: 15 March 2024 Accepted: 20 March 2024 Published: 27 March 2024
  • MSC : 62A86

  • This paper presents a novel methodology aimed at generating chi-square variates within the framework of neutrosophic statistics. It introduces algorithms designed for the generation of neutrosophic random chi-square variates and illustrates the distribution of these variates across a spectrum of indeterminacy levels. The investigation delves into the influence of indeterminacy on random numbers, revealing a significant impact across various degrees of freedom. Notably, the analysis of random variate tables demonstrates a consistent decrease in neutrosophic random variates as the degree of indeterminacy escalates across all degrees of freedom values. These findings underscore the pronounced effect of uncertainty on chi-square data generation. The proposed algorithm offers a valuable tool for generating data under conditions of uncertainty, particularly in scenarios where capturing real data proves challenging. Furthermore, the data generated through this approach holds utility in goodness-of-fit tests and assessments of variance homogeneity.

    Citation: Muhammad Aslam, Osama H. Arif. Simulating chi-square data through algorithms in the presence of uncertainty[J]. AIMS Mathematics, 2024, 9(5): 12043-12056. doi: 10.3934/math.2024588

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  • This paper presents a novel methodology aimed at generating chi-square variates within the framework of neutrosophic statistics. It introduces algorithms designed for the generation of neutrosophic random chi-square variates and illustrates the distribution of these variates across a spectrum of indeterminacy levels. The investigation delves into the influence of indeterminacy on random numbers, revealing a significant impact across various degrees of freedom. Notably, the analysis of random variate tables demonstrates a consistent decrease in neutrosophic random variates as the degree of indeterminacy escalates across all degrees of freedom values. These findings underscore the pronounced effect of uncertainty on chi-square data generation. The proposed algorithm offers a valuable tool for generating data under conditions of uncertainty, particularly in scenarios where capturing real data proves challenging. Furthermore, the data generated through this approach holds utility in goodness-of-fit tests and assessments of variance homogeneity.



    A set CR is said to be convex, if

    (1ϑ)ς1+ϑς2C,ς1,ς2C,ϑ[0,1].

    A mapping ˉΞ:CR is said to be convex, if

    ˉΞ((1ϑ)ς1+ϑς2)(1ϑ)ˉΞ(ς1)+ϑˉΞ(ς2),ς1,ς2C,ϑ[0,1].

    Convex analysis is a vital and extensive branch of mathematics that investigates convex sets and mappings defined over them, as well as their properties. It encompasses both geometric and classical aspects of the problem and has numerous applications in functional analysis, topology, optimization theory, fixed-point theory, economics, engineering, and other fields. The theory of convex mappings is closely linked with the theory of inequalities. Inequalities are crucial in mathematical analysis due to their vast applications. In recent years, the theory of inequalities has attracted many researchers, as it presents various intriguing problems. The development of this theory, in conjunction with convex mapping, has been remarkable over the past few decades. Numerous inequalities can be derived directly from the application of convex mappings. One of the most widely studied results in this context is the Hermite-Hadamard (also known as trapezium) inequality, which provides a necessary and sufficient condition for a mapping to be convex. It reads as:

    Let ˉΞ:I=[ς1,ς2]RR be a convex mapping, then

    ˉΞ(ς1+ς22)1ς2ς1ς2ς1ˉΞ(x)dxˉΞ(ς1)+ˉΞ(ς2)2.

    One of the fruitful results pertaining to the convexity property of the mappings is Jensen's inequality, which reads as:

    Theorem 1.1. [1] Let 0<x1x2x3.xn and let μ=(μ1,μ2,,μn) nonnegative weights such that ni=0μ=1. If ˉΞ:I=[ς1,ς2]RR is a convex mapping, then

    ˉΞ(ni=1μixi)ni=1μiˉΞ(xi),

    where xi[ς1,ς2] and μi[0,1], (i=¯1,n).

    In 2001 Mercer, [2] derived the discrete version of the Jensen inequality known as the Jensen-Mercer inequality, and is given as:

    Theorem 1.2. Let ˉΞ:I=[ς1,ς2]RR be a convex mapping, then

    ˉΞ(ς1+ς2ni=1μixi)ˉΞ(ς1)+ˉΞ(ς2)ni=1μiˉΞ(xi), (1.1)

    for each xi[ς1,ς2] and μi[0,1], (i=¯1,n) with ni=1μ=1.

    Convex mapping theory has numerous applications in approximation theory, particularly in obtaining error bounds using inequalities. The Hermite-Hadamard inequality (HHI) plays a crucial role in some trapezoidal quadrature rules and estimations. Specifically, the left and right estimations of HHI provide the error bounds for the midpoint rule and trapezoidal rule, respectively. For details, see [3].

    Building upon inequality (1.1), Kian and Moslehian [4] proposed a new refinement and extension of the trapezium inequality. Subsequently, Ogulmus et al. [5] derived fractional analogues of TJM-like inequalities based on their work. There have been several recent publications on this inequality. For example, see [6,7,8,9,10].

    The h and q calculus was introduced by Euler in the 18th century, and since then has undergone significant developments and gained much attention. It is a type of time scale calculus that focuses on the 0<q<1 domain, with major areas of study including q-derivatives, q-integrals, and their generalizations using various techniques such as pq-calculus, interval valued technique, and trapezoidal strips. This calculus has numerous applications in special mappings, physics, number theory, combinatorics, cryptography, and other fields.

    In recent years, the study of inequalities based on q-calculus has become a prominent subject in mathematical analysis. Many researchers have dedicated their efforts to obtaining new q-analogues of classical inequalities. For example, Sudsutad et al. [11] discussed q-Hölder's inequality and Hermite-Hadamard's inequalities via quantum calculus, while Noor et al. [12] derived some q-variants of integral inequalities. In 2018, Alp et al. [13] corrected the q-HHI derived some q-mid-point type inequalities. Other recent developments include Zhang et al. [14] formulating q-integral inequalities via (α,m) convex mappings, Deng et al. [15] investigating a stronger version of q-integral inequalities in terms of preinvex mappings, Kunt [16] deriving fractional quantum variants of HHI, and Cortez et al. [17] deriving fractional quantum integral inequalities involving new generalized convex mappings.

    Further research in this field has resulted in Iftikhar et al. [18] proposing new quantum analogies of Simpson's type inequalities, Wang [19] presenting some q-outcomes related to s-preinvexity, Chu et al. [20] introducing the concept of generalized right q-derivatives and integrals and developing new Ostrowski's type inequalities involving n-polynomial convex mappings, and authors in [21] deducing some parametric quantum inequalities with respect to preinvex mappings. Additionally, Kalsoom et al. [22] analyzed the generalized quantum integral inequalities involving preinvex mappings, Ali et al. [23] incorporated with some q-mid-point HHI and derived some trapezoidal type inequalities, and Bin-Mohsin et al. [24] established some new generalized TJM type inequalities in the light of q-calculus. The quantum and post quantum variants of TJM inequalities have also been proven in [25,26], which has opened new avenues for researchers. For more information, refer to [27,28,29].

    Our paper aims to explore the mid-point-TJM inequality by utilizing q-concepts and majorization techniques. The theory of majorization plays a significant role in mathematical analysis and has a wide range of applications in information theory and inequality theory. It encompasses both discrete and continuous forms of inequalities, which are detailed in [30,31,32]. Our work is organized as follows: the first section serves as an introduction to the topic, while the subsequent section provides a review of essential definitions and facts that are instrumental in proving our main results. The third section discusses quantum TJM inequalities and presents two new q-integral identities in the first subsection, followed by the presentation of new associated bounds in the third subsection. In the final section, we delve into the applications of our findings, including numerical examples and graphical analysis. The novelty of the current study is that we develop some new generalized TJM inequalities in the frame of quantum calculus by linking the concepts of majorization theory. Results obtained in the current study unifies many known and new results in both continuous and discrete form. We hope that our approach and methods will stimulate further research in this field.

    In this section, we discuss some preliminaries which will be helpful during the study of this paper. First of all, we recall some basic concepts from quantum calculus. Tariboon and Ntouyas have defined the q-derivative as:

    Definition 2.1. [33] Assume ˉΞ:J=[ς1,ς2]RR is a continuous mapping and suppose uJ, then

    ς1DqˉΞ(u)=ˉΞ(u)ˉΞ(qu+(1q)ς1)(1q)(uς1),uς1,0<q<1. (2.1)

    We say that ˉΞ is q-differentiable on J provided ς1DqˉΞ(u) exists for all uJ. Note that if ς1=0 in (2.1), then 0DqˉΞ=DqˉΞ, where Dq is the well-known classical q -derivative of the mapping ˉΞ(u) defined by

    DqˉΞ(u)=ˉΞ(u)ˉΞ(qu)(1q)u.

    Also, here and further we use the following notation for q-number

    [n]q=1qn1q=1+q+q2++qn1,q(0,1).

    Jackson gave the definition of the q-Jackson integral from 0 to ς2 for 0<q<1 as follows:

    ς20ˉΞ(ϱ)0dqϱ=(1q)ς2n=0qnˉΞ(bqn) (2.2)

    provided the sum converges absolutely. Jackson also gave the q-Jackson integral on a generic interval [ς1,ς2] as:

    ς2ς1ˉΞ(ϱ)dqϱ=ς20ˉΞ(ϱ)dqϱ+ς10ˉΞ(ϱ)dqϱ.

    We now give the definition of the qς1-definite integral.

    Definition 2.2. [33] Let ˉΞ:[ς1,ς2]R be a continuous mapping. Then, the qς1-definite integral on [ς1,ς2] is defined as:

    ς2ς1ˉΞ(ϱ)ς1dqϱ=(1q)(ς2ς1)n=0qnˉΞ(qnς2+(1qn)ς1)=(ς2ς1)10ˉΞ((1ϱ)ς1+ϱς2)dqϱ.

    Now we recall some more important results, which will help us in deriving our main result.

    Theorem 2.1. [33] If ˉΞ:[ς1,ς2]R is a CM and u[ς1,ς2], then

    ς1Dqzς1ˉΞ(u)ς1dqu=ˉΞ(z).zcς1DqˉΞ(u)ς1dqu=ˉΞ(z)ˉΞ(c).

    Now we recall the definitions of the qς2 derivative and definite integrals.

    Definition 2.3. [34] Let ˉΞ:[ς1,ς2]R be a CM and u[ς1,ς2], then

    bDqˉΞ(u)=ˉΞ(qu+(1q)ς2)ˉΞ(u)(1q)(ς2u),u<ς2.

    Definition 2.4. [34] Let ˉΞ:[ς1,ς2]R be a continuous mapping. Then, the qς2-definite integral on [ς1,ς2] is defined as:

    ς2ς1ˉΞ(ϱ)ς2dqϱ=(1q)(ς2ς1)n=0qnˉΞ(qnς1+(1qn)ς2)=(ς2ς1)10ˉΞ(ϑς1+(1ϱ)ς2)dqϱ.

    Now we rewrite some further results.

    Theorem 2.2. [34] If ˉΞ:[ς1,ς2]R is a CM and u[ς1,ς2], then

    ς2Dqς2zˉΞ(u)ς2dqu=ˉΞ(z).ς2zς2DqˉΞ(u)ς2dqu=ˉΞ(ς2)ˉΞ(z).

    Lemma 2.1. [34] For a continuous mappings ˉΞ,Φ:[ς1,ς2]R, then

    c0Φ(ϑ)ς2DqˉΞ(ϑς1+(1ϑ)ς2)dqϑ=1ς2ς1c0DqΦ(ϑ)ˉΞ(qϑς1+(1q_ϑ)ς2)dqϑΦ(ϑ)ˉΞ(ϑς1+(1ϑ)ς2)ς2ς1|c0.

    Using the Definition 2.4, one can have the following quantum version of the Hermite-Hadamard's inequality.

    Theorem 2.3. [13] Let ˉΞ:[ς1,ς2]R be a convex mapping, then for 0<q<1, we have

    ˉΞ(ς1+qς21+q)1ς2ς1ς2ς1ˉΞ(u)ς2dquˉΞ(ς1)+qˉΞ(ς2)1+q. (2.3)

    Inspired by ongoing research, Ali et al. [23] calculated a new quantum version of the mid-point trapezium inequality.

    Theorem 2.4. [23] Let ˉΞ:[ς1,ς2]R be a convex mapping then

    ˉΞ(ς1+ς22)1ς2ς1[ς1+ς22ς1ˉΞ(x)ς1dqx+ς2ς1+ς22ˉΞ(x)ς2dqx]ˉΞ(ς1)+ˉΞ(ς2)2. (2.4)

    Now we recall some known definitions and results regarding majorization theory.

    Definition 2.5. [35] Let ς1=(ς11,ς12,,ς1l) and ς2=(ς21,ς22,,ς2l) be two l-tuples of real numbers and ς1[1]ς1[2]ς2[l],ς2[1]ς2[2]ς2[l] be their ordered components, then ς1 is said to majorize ς2 (symbolically ς2ς1), if

    ks=1ς2[s]ks=1ς1[s]k=1,2,3,,l1

    and

    ls=1ς2[s]=ls=1ς1[s].

    Majorization is a partial ordered relation of two l-tuples ς1=(ς11,ς12,,ς1l) and ς2=(ς21,ς22,,ς2l) which explains that ς1 is more nearly equal to ς2. Now we recall majorization theorem due to Hardy, Littlewood and Polya [36].

    Theorem 2.5. Let ς1=(ς11,ς12,,ς1l) and ς2=(ς21,ς22,,ς2l) be two real l-tuples such that ς1s,ς2sI=[ς1,ς2]. Then

    ls=1f(ς2s)ls=1f(ς1s)

    is valid for each continuous convex mapping ˉΞ:IR if and only if ς2ς1.

    The weighted version of the above theorem is given as:

    Theorem 2.6. [37] Let ˉΞ:IR be a continuous convex mapping and ς1=(ς11,ς12,,ς1l),ς2=(ς21,ς22,,ς2l) and p=(p1,p2,,pl) be the three l-tuples such that ς1s,ς2sI,ps0,s{1,2,3,.,l}. If ς2 is a decreasing l-tuple, then

    ks=1psς2[s]ks=1psς1[s]k=1,2,3,,l1, (2.5)
    ls=1psς2[s]=ls=1psς1[s],

    then

    ls=1psˉΞ(ς2s)ls=1psˉΞ(ς1s).

    Theorem 2.7. [38] Suppose that ˉΞ:[ς1,ς2]R be a real valued convex mapping (xij) is a n×m real matrix and u=(u1,u2,,ul) is a l-tuple such that uj,xij,wi0 for i=1,2,3,,n with ni=1wi=1. If u every row of xij then

    ˉΞ(lj=1usl1j=1ni=1wixij)lj=1ˉΞ(us)l1j=1ni=1wif(xij).

    In this section, we derive q-TJM inequality associated with convex mapping. Furthermore, we also establish q-TJM like inequalities with the help of auxiliary results.

    In the following section, we established a new mid-point TJM- inequality involving convexity and q-integrability of ˉΞ.

    Theorem 3.1. Suppose that ˉΞ:I=[ς1,ς2]R is real valued convex mapping and ϖ=(ϖ1,ϖ2,,ϖl),ξ=(ξ1,ξ2,,ξl),η=(η1,η2,,ηl) are three l-tuples ϖs,ξs,ηs for all s{1,2,3,,l}. If ξϖ and ηϖ, then

    ˉΞ(ls=1ϖsl1s=1ξs+ηs2)ls=1ˉΞ(ϖs)l1s=11ηsξs[ξs+ηs2ξsˉΞ(u)ξsdqu+ηsξs+ηs2ˉΞ(u)ηsdqu]ls=1ˉΞ(ϖs)l1s=1ˉΞ(ξs+ηs2) (3.1)

    and

    ˉΞ(ls=1ϖsl1s=1ξs+ηs2)1l1s=1ηsξs[ls=1ϖsl1s=1ξs+ηs2ls=1ϖsl1s=1ηsˉΞ(u)ls=1ϖsl1s=1ηsdqu+ls=1ϖsl1s=1ξsls=1ϖsl1s=1ξs+ηs2ˉΞ(u)ls=1ϖsl1s=1ξsdqu]ls=1ˉΞ(ϖs)l1s=1ˉΞ(ξs)+ˉΞ(ηs)2. (3.2)

    Proof. To use Theorem 2.7 we show that ϖ majorizes r and z where r=(r1,r2,,rl),z=(z1,z2,,zl),rj=ϑ2ξs+(2ϑ)2ηs and zj=ϑ2ηs+(2ϑ)2ξs for s={1,2,3,,l}.

    For this, let kj=1ξ[j]=β1k and kj=1η[j]=β2k for k=1,2,,l1. Then

    kj=1r[j]=ϑ2kj=1ξ[j]+(2ϑ)2kj=1η[j]=ϑ2β1k+(2ϑ)2β2k.

    Since ξϖ and ηϖ then by definition of majorization, we have kj=1ξ[j]kj=1ϖ[j] and kj=1η[j]kj=1ϖ[j] that is

    β1kkj=1ϖ[j] (3.3)

    and

    β2kkj=1ϖ[j]. (3.4)

    Multiplying (3.3) by ϑ2 and (3.4) by 2ϑ2 and then adding the resulting inequalities, we get

    kj=1r[j]=ϑ2β1k+(2ϑ)2β2kkj=1ϖ[j]. (3.5)

    But ls=1ϖs=ls=1ξs and ls=1ϖs=ls=1ηs, then by using (3.5), we have

    ls=1rs=ls=1ϖs.

    Hence rϖ. Similarly, we can show that zϖ. Then by using Theorem 3.2 for w1=w2=12, we have

    ˉΞ(ls=1ϖsl1s=1ξs+ηs2)ls=1ˉΞ(ϖs)l1s=112[ˉΞ(ϑ2ξs+2ϑ2ηs)+ˉΞ(ϑ2ηs+2ϑ2ξs)]. (3.6)

    Now taking the q-integration of (3.6), we have

    ˉΞ(ls=1ϖsl1s=1ξs+ηs2)ls=1ˉΞ(ϖs)l1s=112[10ˉΞ(ϑ2ξs+2ϑ2ηs)dqϑ+10ˉΞ(ϑ2ηs+2ϑ2ξs)dqϑ].

    After simplifying, we obtain the left inequality of (3.1).

    For the right inequality of (3.1), using (2.4), we have

    l1s=11ηsξs[ξs+ηs2ξsˉΞ(u)ξsdqu+ηsξs+ηs2ˉΞ(u)ηsdqu]l1s=1ˉΞ(ξs+ηs2). (3.7)

    Adding l1s=1ˉΞ(ϖs) on both sides of (3.7), we obtain our required result. In this way, we complete the proof of our (3.1).

    To prove (3.2), we apply the definition of convex mapping, we have

    ˉΞ(ls=1ϖsl1s=1ξs+ηs2)=ˉΞ[12{(ϑ2(ls=1ϖsl1s=1ηs)+2ϑ2(ls=1ϖsl1s=1ξs))+(ϑ2(ls=1ϖsl1s=1ξs)+2ϑ2(ls=1ϖsl1s=1ηs))}]2ˉΞ(ls=1ϖsl1s=1ξs+ηs2)ˉΞ((1ϑ)(ls=1ϖsl1s=1ξs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))+ˉΞ((1ϑ)(ls=1ϖsl1s=1ηs)+ϑ(ls=1ϖsl1s=1ξs+ηs2)). (3.8)

    Now, taking the q-integration of (3.8) on both sides over [0,1] and using the definitions (2.1) and (2.3), then

    2ˉΞ(ls=1ϖsl1s=1ξs+ηs2)10ˉΞ((1ϑ)(ls=1ϖsl1s=1ξs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))dqϑ+10ˉΞ((1ϑ)(ls=1ϖsl1s=1ηs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))dqϑ.=2l1s=1ηsξs[ls=1ϖsl1s=1ξs+ηs2ls=1ϖsl1s=1ηsˉΞ(u)ls=1ϖsl1s=1ηsdqu+ls=1ϖsl1s=1ξsls=1ϖsl1s=1ξs+ηs2ˉΞ(u)ls=1ϖsl1s=1ξsdqu].

    Hence we complete the proof of our first inequality of (3.2).

    To prove, our second inequality, we use the notion of convex mappings,

    ˉΞ(ls=1ϖsϑ2l1s=1ξs2ϑ2l1s=1ηs)ls=1ˉΞ(ϖs)ϑ2l1s=1ˉΞ(ξs)2ϑ2l1s=1ˉΞ(ηs). (3.9)
    ˉΞ(ls=1ϖs2ϑ2l1s=1ξsϑ2l1s=1ηs)ls=1ˉΞ(ϖs)ϑ2l1s=1ˉΞ(ηs)2ϑ2l1s=1ˉΞ(ξs). (3.10)

    Adding (3.9) and (3.10), then q-integration of resulting inequality yields the required inequality.

    If we choose l=2, then

    ˉΞ(ϖ1+ϖ2ηs+ξs2)ˉΞ(ϖ1)+ˉΞ(ϖ2)1η1ξ1[ξ1+η12ξ1ˉΞ(u)ξ1dqu+η1ξ1+η12ˉΞ(u)η1dqu]ˉΞ(ϖ1)+ˉΞ(ϖ2)ˉΞ(η1+ξ12)

    and

    ˉΞ(ϖ1+ϖ2ξ1+η12)1η1ξ1[ϖ1+ϖ2ξ1+η12ϖ1+ϖ2η1ˉΞ(u)ϖ1+ϖ2η1dqu+ϖ1+ϖ2ξ1ϖ1+ϖ2ξ1+η12ˉΞ(u)ϖ1+ϖ2ξ1dqu]ˉΞ(ϖ1)+ˉΞ(ϖ2)ˉΞ(ξ1)+ˉΞ(η1)2.

    For further demonstration, we discuss a numeric example in the support of Theorem 3.1.

    Example 3.1. Considering ˉΞ(u)=u2, with ϖ1=1,ξ1=1,η1=2,ϖ2=3 and q=0.5, then

    ˉΞ(ϖ1+ϖ2ξ1+η12)=(12)2=14.ϖ1+ϖ2ξ1+η12ϖ1+ϖ2η1ˉΞ(u)ϖ1+ϖ2η1dqu=120u20dqu=114.ϖ1+ϖ2ξ1ϖ1+ϖ2ξ1+η12ˉΞ(u)ϖ1+ϖ2ξ1dqu=112u21dqu=521.ˉΞ(ϖ1)+ˉΞ(ϖ2)ˉΞ(ξ1)+ˉΞ(η1)2=152

    From above calculations, we can infer that 0.25<0.31<7.5.

    In the following subsection, our first target is to derive two new quantum integral identities involving ς1q and ς2q differentiability of the mappings and ordered n-tuples. Here, we propose a new general identity of mid-point type which will play a critical role in order to compute some new error bounds of mid-point schemes.

    Lemma 3.1. Let ϖ=(ϖ1,ϖ2,ϖ3,,ϖl), ξ=(ξ1,ξ2,..,ξl) and η=(η1,η2,,ηl) be the three l-tuples such that ϖs,ξs,ηs[I] for all s{1,2,,l},ϑ[0,1] and ˉΞ:JR be a CM and 0<q<1. If ls=1ϖsl1s=1ξsDqˉΞ and ls=1ϖsl1s=1ηsDqˉΞ an integrable mapping on J, then

    ϑ(ϖs;ξs;ηs)=l1s=1(ηsξs)4[10qϑls=1ϖsl1s=1ξsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ξs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))dqϑ10qϑls=1ϖsl1s=1ηsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ηs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))dqϑ], (3.11)

    where

    ϑ(ϖs;ξs;ηs):=1l1s=1(ηsξs)[ls=1ϖsl1s=1ξs+ηs2ls=1ϖsl1s=1ηsˉΞ(u)ls=1ϖsl1s=1ηsdqu+ls=1ϖsl1s=1ξsls=1ϖsl1s=1ξs+ηs2ˉΞ(u)ls=1ϖsl1s=1ξsdqu]ˉΞ(ls=1ϖsl1s=1ξs+ηs2).

    Proof. Consider the right-hand side of (3.11) as

    I=l1s=1(ηsξs)4[I1I2]. (3.12)

    By Lemma 2.1, we have

    I1=10qϑls=1ϖsl1s=1ξsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ξs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))dqϑ=2ql1s=1(ηsξs)10ˉΞ((1qϑ)(ls=1ϖsl1s=1ξs)+qϑ(ls=1ϖsl1s=1ξs+ηs2))0dqϑ2ql1s=1(ηsξs)ˉΞ(ls=1ϖsl1s=1ξs+ηs2)=2(1q)l1s=1(ηsξs)n=0qn+1ˉΞ((1qn+1)(ls=1ϖsl1s=1ξs)+qn+1(ls=1ϖsl1s=1ξs+ηs2))2ql1s=1(ηsξs)ˉΞ(ls=1ϖsl1s=1ξs+ηs2)=2(1q)l1s=1(ηsξs)n=1qnˉΞ((1qn)(ls=1ϖsl1s=1ξs)+qn(ls=1ϖsl1s=1ξs+ηs2))2ql1s=1(ηsξs)ˉΞ(ls=1ϖsl1s=1ξs+ηs2)=2(1q)l1s=1(ηsξs)n=0qnˉΞ((1qn)(ls=1ϖsl1s=1ξs)+qn(ls=1ϖsl1s=1ξs+ηs2))2l1s=1(ηsξs)ˉΞ(ls=1ϖsl1s=1ξs+ηs2)=4(l1s=1(ηsξs))2ls=1ϖsl1s=1ξsls=1ϖsl1s=1ξs+ηs2ˉΞ(u)ls=1ϖsl1s=1ξsdqu2l1s=1(ηsξs)ˉΞ(ls=1ϖsl1s=1ξs+ηs2). (3.13)

    Similarly, we get

    I2=10qϑls=1ϖsl1s=1ηsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ηs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))dqϑ=4(l1s=1(ηsξs))2ls=1ϖsl1s=1ξs+ηs2ls=1ϖsl1s=1ηsˉΞ(u)ls=1ϖsl1s=1ηsdqu2l1s=1(ηsξs)ˉΞ(ls=1ϖsl1s=1ξs+ηs2). (3.14)

    Combination of (3.12)–(3.14) yields the required result.

    If we choose l=2 in Lemma 3.1, then we have

    1η1ξ1[ϖ1+ϖ2ξ1+η12ϖ1+ϖ2η1ˉΞ(u)ϖ1+ϖ2η1dqu+ϖ1+ϖ2ξ1ϖ1+ϖ2ξ1+η12ˉΞ(u)ϖ1+ϖ2ξ1dqu]ˉΞ(ϖ1+ϖ2ξ1+η12)=η1ξ14[10qϑϖ1+ϖ2ξ1DqˉΞ((1ϑ)(ϖ1+ϖ2ξ1)+ϑ(ϖ1+ϖ2ξ1+η12))dqϑ10qϑϖ1+ϖ2η1DqˉΞ((1ϑ)(ϖ1+ϖ2η1)+ϑ(ϖ1+ϖ2ξ1+η12))dqϑ].

    Now, we establish a general Dragomir-Agarwal-type inequality, which is crucial to establish some error bounds of the trapezoidal rule.

    Lemma 3.2. Let ϖ=(ϖ1,ϖ2,ϖ3,,ϖl), ξ=(ξ1,ξ2,..,ξl) and η=(η1,η2,,ηl) be the three l-tuples such that ϖs,ξs,ηs[I] for all s{1,2,,l},ϑ[0,1] and ˉΞ:JR be a CM and 0<q<1. If ls=1ϖsl1s=1ξsDqˉΞ and ls=1ϖsl1s=1ηsDqˉΞ an integrable mapping on J, then

    Ω(ϖs;ηs;ξs)=l1s=1(ηsξs)4[10(1qϑ)ls=1ϖsl1s=1ξsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ξs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))dqϑ+10(qϑ1)ls=1ϖsl1s=1ηsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ηs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))dqϑ]. (3.15)

    where

    Ω(ϖs;ηs;ξs)=:ˉΞ(ls=1ϖsl1s=1ξs)+ˉΞ(ls=1ϖsl1s=1ηs)21l1s=1(ηsξs)[ls=1ϖsl1s=1ξs+ηs2ls=1ϖsl1s=1ηsˉΞ(u)ls=1ϖsl1s=1ηsdqu+ls=1ϖsl1s=1ξsls=1ϖsl1s=1ξs+ηs2ˉΞ(u)ls=1ϖsl1s=1ξsdqu].

    Proof. Consider the right-hand side of (3.15) as

    J=l1s=1(ηsξs)4[J1+J2]. (3.16)

    By Lemma 2.1, we have

    J1=10(1qϑ)ls=1ϖsl1s=1ξsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ξs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))dqϑ=2ql1s=1(ηsξs)10ˉΞ((1qϑ)(ls=1ϖsl1s=1ξs)+qϑ(ls=1ϖsl1s=1ξs+ηs2))0dqϑ2(1q)l1s=1(ηsξs)ˉΞ(ls=1ϖsl1s=1ξs+ηs2)+2l1s=1(ηsξs)ˉΞ(ls=1ϖsl1s=1ξs)=2(1q)l1s=1(ηsξs)n=0qn+1ˉΞ((1qn+1)(ls=1ϖsl1s=1ξs)+qn+1(ls=1ϖsl1s=1ξs+ηs2))2(1q)l1s=1(ηsξs)ˉΞ(ls=1ϖsl1s=1ξs+ηs2)+2l1s=1(ηsξs)ˉΞ(ls=1ϖsl1s=1ξs)=2(1q)l1s=1(ηsξs)n=1qnˉΞ((1qn)(ls=1ϖsl1s=1ξs)+qn(ls=1ϖsl1s=1ξs+ηs2))2(1q)ql1s=1(ηsξs)ˉΞ(ls=1ϖsl1s=1ξs+ηs2)+2l1s=1(ηsξs)ˉΞ(ls=1ϖsl1s=1ξs)=2(1q)l1s=1(ηsξs)n=0qnˉΞ((1qn)(ls=1ϖsl1s=1ξs)+qn(ls=1ϖsl1s=1ξs+ηs2))+2l1s=1(ηsξs)ˉΞ(ls=1ϖsl1s=1ξs)=4(l1s=1(ηsξs))2ls=1ϖsl1s=1ξsls=1ϖsl1s=1ξs+ηs2ˉΞ(u)ls=1ϖsl1s=1ξsdqu+2l1s=1(ηsξs)ˉΞ(ls=1ϖsl1s=1ξs). (3.17)

    Similarly, we get

    J2=10(qϑ1)ls=1ϖsl1s=1ηsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ηs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))dqϑ=2l1s=1(ηsξs)ˉΞ(ls=1ϖsl1s=1ηs)4(l1s=1(ηsξs))2ls=1ϖsl1s=1ξs+ηs2ls=1ϖsl1s=1ηsˉΞ(u)ls=1ϖsl1s=1ηsdqu. (3.18)

    Comparing (3.16)–(3.18), we conclude our required result.

    If we choose l=2 in Lemma 3.2, then

    ˉΞ(ϖ1+ϖ2ξ1)+ˉΞ(ϖ1+ϖ2η1)21η1ξ1[ϖ1+ϖ2ξ1+η12ϖ1+ϖ2η1ˉΞ(u)ϖ1+ϖ2η1dqu+ϖ1+ϖ2ξ1ϖ1+ϖ2ξ1+η12ˉΞ(u)ϖ1+ϖ2ξ1dqu]=η1ξ14[10(1qϑ)ϖ1+ϖ2ξ1DqˉΞ((1ϑ)(ϖ1+ϖ2ξ1)+ϑ(ϖ1+ϖ2ξ1+η12))dqϑ+10(qϑ1)ϖ1+ϖ2η1DqˉΞ((1ϑ)(ϖ1+ϖ2η1)+ϑ(ϖ1+ϖ2ξ1+η12))dqϑ].

    In this section, we propose some new generalized left and right estimations connecting to newly proved q-TJM inequality proved in the previous section. We use auxiliary results, the convexity property of the mappings, and some well-known inequalities to obtain new refinements of existing results.

    Theorem 3.2. Under the assumptions of Lemma 3.1 and if |ls=1ϖsl1s=1ξsDq| and |ls=1ϖsl1s=1ηsDq| are convex mappings, then

    |ϑ(ϖs;ηs;ξs)|l1s=1(ηsξs)4[q[2]qls=1|ls=1ϖsl1s=1ξsDqˉΞ(ϖs)|q[[3]q+q2]2[2]q[3]ql1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ξs)|q2[3]ql1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ηs)|+q[2]qls=1|ls=1ϖsl1s=1ηsDqˉΞ(ϖs)|q2[3]ql1s=1|ls=1ϖsl1s=1ηsDqˉΞ(ξs)|q[[3]q+q2]2[2]q[3]ql1s=1|ls=1ϖsl1s=1ηsDqˉΞ(ηs)|].

    Proof. Using Lemma 3.1, property of modulus and the convexity of |ls=1ϖsl1s=1ξsDq| and |ls=1ϖsl1s=1ηsDq|, we have

    |ϑ(ϖ;η;ξ)|l1s=1(ηsξs)4[10qϑ|ls=1ϖsl1s=1ξsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ξs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))|dqϑ10qϑ|ls=1ϖsl1s=1ηsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ηs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))|dqϑ]l1s=1(ηsξs)4[10qϑ(ls=1|ls=1ϖsl1s=1ξsDqˉΞ(ϖs)|2ϑ2l1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ξs)|ϑ2l1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ηs)|)dqϑ]+l1s=1(ηsξs)4[10qϑ(ls=1|ls=1ϖsl1s=1ηsDqˉΞ(ϖs)|ϑ2l1s=1|ls=1ϖsl1s=1ηsDqˉΞ(ξs)|2ϑ2l1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ηs)|)dqϑ].

    After simple calculations, we achieve our final result.

    If we choose l=2 in Theorem 3.2, then

    |1η1ξ1[ϖ1+ϖ2ξ1+η12ϖ1+ϖ2η1ˉΞ(u)ϖ1+ϖ2η1dqu+ϖ1+ϖ2ξ1ϖ1+ϖ2ξ1+η12ˉΞ(u)ϖ1+ϖ2ξ1dqu]ˉΞ(ϖ1+ϖ2ξ1+η12)|η1ξ14[q[2]q(|ϖ1+ϖ2ξ1DqˉΞ(ϖ1)|+|ϖ1+ϖ2ξ1DqˉΞ(ϖ2)|)q[[3]q+q2]2[2]q[3]q|ϖ1+ϖ2ξ1DqˉΞ(ξ1)|q2[3]q|ϖ1+ϖ2ξ1DqˉΞ(η1)|+q[2]q(|ϖ1+ϖ2η1DqˉΞ(ϖ1)|+|ϖ1+ϖ2η1DqˉΞ(ϖ2)|)q2[3]q|ϖ1+ϖ2η1DqˉΞ(ξ1)|q[[3]q+q2]2[2]q[3]q|ϖ1+ϖ2η1DqˉΞ(η1)|].

    Theorem 3.3. Under the assumptions of Lemma 3.1 and if |ls=1ϖsl1s=1ξsDq|r and |ls=1ϖsl1s=1ηsDq|r are convex mappings with 1r+1s=1, then we have

    |ϑ(ϖ;η;ξ)|l1s=1(ηsξs)4(q[2]q)11r[(q[2]q(ls=1|ls=1ϖsl1s=1ξsDqˉΞ(ϖs)|r)q[[3]q+q2]2[2]q[3]ql1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ξs)|rq2[3]ql1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ηs)|r)1r+(q[2]q(ls=1|ls=1ϖsl1s=1ηsDqˉΞ(ϖs)|r)q2[3]ql1s=1|ls=1ϖsl1s=1ηsDqˉΞ(ξs)|rq[[3]q+q2]2[2]q[3]ql1s=1|ls=1ϖsl1s=1ηsDqˉΞ(ηs)|r)1r].

    Proof. Using Lemma 3.1, property of modulus, power-mean inequality and the convexity property of |ls=1ϖsl1s=1ξsDq|r and |ls=1ϖsl1s=1ηsDq|r with 1r+1s=1, we have

    |ϑ(ϖ;η;ξ)|l1s=1(ηsξs)4(10qϑdqϑ)11r[(10qϑ|ls=1ϖsl1s=1ξsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ξs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))|rdqϑ)1r+(10qϑ|ls=1ϖsl1s=1ηsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ηs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))|rdqϑ)1r]l1s=1(ηsξs)4(q[2]q)1r[(10qϑ(ls=1|ls=1ϖsl1s=1ξsDqˉΞ(ϖs)|r2ϑ2l1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ξs)|rϑ2l1s=1l1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ηs)|r)dqϑ)1r+(10qϑ(ls=1|ls=1ϖsl1s=1ηsDqˉΞ(ϖs)|rϑ2l1s=1|ls=1ϖsl1s=1ηsDqˉΞ(ξs)|r2ϑ2l1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ηs)|r)dqϑ)1r].

    After simple calculations, we achieve our final result.

    If we choose l=2 in Theorem 3.3, then

    |1η1ξ1[ϖ1+ϖ2ξ1+η12ϖ1+ϖ2η1ˉΞ(u)ϖ1+ϖ2η1dqu+ϖ1+ϖ2ξ1ϖ1+ϖ2ξ1+η12ˉΞ(u)ϖ1+ϖ2ξ1dqu]ˉΞ(ϖ1+ϖ2ξ1+η12)|η1ξ14(q[2]q)11r[(q[2]q(|ϖ1+ϖ2ξ1DqˉΞ(ϖ1)|r+|ϖ1+ϖ2ξ1DqˉΞ(ϖ2)|r)q[[3]q+q2]2[2]q[3]q|ϖ1+ϖ2ξ1DqˉΞ(ξ1)|rq2[3]q|ϖ1+ϖ2ξ1DqˉΞ(η1)|r)1r+(q[2]q(|ϖ1+ϖ2η1DqˉΞ(ϖ1)|r+|ϖ1+ϖ2η1DqˉΞ(ϖ2)|r)q2[3]q|ϖ1+ϖ2η1DqˉΞ(ξ1)|rq[[3]q+q2]2[2]q[3]q|ϖ1+ϖ2η1DqˉΞ(η1)|r)1r].

    Theorem 3.4. Under the assumptions of Lemma 3.1 and if |ls=1ϖsl1s=1ξsDq|r and |ls=1ϖsl1s=1ηsDq|r are convex mappings with 1r+1s=1, then we have

    |ϑ(ϖs;ηs;ξs)|l1s=1(ηsξs)4(qr[r+1]q)11r[(ls=1|ls=1ϖsl1s=1ξsDqˉΞ(ϖs)|r1+2q2[2]ql1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ξs)|r12[2]ql1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ηs)|r)1r+(ls=1|ls=1ϖsl1s=1ηsDqˉΞ(ϖs)|r12[2]ql1s=1|ls=1ϖsl1s=1ηsDqˉΞ(ξs)|r1+2q2[2]ql1s=1|ls=1ϖsl1s=1ηsDqˉΞ(ηs)|r)1r].

    Proof. Using Lemma 3.1, property of modulus, Hölder's inequality and using the convexity property of |ls=1ϖsl1s=1ξsDq|r and |ls=1ϖsl1s=1ηsDq|r with 1r+1s=1, we have

    |ϑ(ϖs;ηs;ξs)|l1s=1(ηsξs)4(10(qϑ)rdqϑ)11r×[(10|ls=1ϖsl1s=1ξsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ξs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))|rdqϑ)1r+(10|ls=1ϖsl1s=1ηsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ηs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))|rdqϑ)1r]l1s=1(ηsξs)4(qr[r+1]pq)1r[(10(ls=1|ls=1ϖsl1s=1ξsDqˉΞ(ϖs)|r2ϑ2l1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ξs)|rϑ2l1s=1l1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ηs)|r)dqϑ)1r+(10(ls=1|ls=1ϖsl1s=1ηsDqˉΞ(ϖs)|rϑ2l1s=1|ls=1ϖsl1s=1ηsDqˉΞ(ξs)|r2ϑ2l1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ηs)|r)dqϑ)1r].

    After simple calculations, we achieve our final result.

    If we choose l=2 in Theorem 3.4, then

    1η1ξ1[ϖ1+ϖ2ξ1+η12ϖ1+ϖ2η1ˉΞ(u)ϖ1+ϖ2η1dqu+ϖ1+ϖ2ξ1ϖ1+ϖ2ξ1+η12ˉΞ(u)ϖ1+ϖ2ξ1dqu]ˉΞ(ϖ1+ϖ2ξ1+η12)η1ξ14(qr[r+1]q)11r[(|ϖ1+ϖ2ξ1DqˉΞ(ϖ1)|r+|ϖ1+ϖ2ξ1DqˉΞ(ϖ2)|r1+2q2[2]q|ϖ1+ϖ2ξ1DqˉΞ(ξ1)|r12[2]q|ϖ1+ϖ2ξ1DqˉΞ(η1)|r)1r+(|ϖ1+ϖ2η1DqˉΞ(ϖ1)|r+|ϖ1+ϖ2η1DqˉΞ(ϖ2)|r12[2]q|ϖ1+ϖ2η1DqˉΞ(ξ1)|r1+2q2[2]q|ϖ1+ϖ2η1DqˉΞ(η1)|r)1r].

    Now we derive some new results related to the right side of TJM inequality using Lemma 3.2.

    Theorem 3.5. Under the assumptions of Lemma 3.2 and if |ls=1ϖsl1s=1ξsDq| and |ls=1ϖsl1s=1ηsDq| are convex mappings, we have

    |Ω(ϖs;ηs;ξs)|l1s=1(ηsξs)4[1[2]q(ls=1|ls=1ϖsl1s=1ξsDqˉΞ(ϖs)|)[2[3]q1]2[2]q[3]q|ls=1ϖsl1s=1ξsDqˉΞ(ξs)|12[2]q[3]q|ls=1ϖsl1s=1ξsDqˉΞ(ηs)|+1[2]q(ls=1|ls=1ϖsl1s=1ηsDqˉΞ(ϖs)|)12[2]q[3]q|ls=1ϖsl1s=1ηsDqˉΞ(ξs)|[2[3]q1]2[2]q[3]q|ls=1ϖsl1s=1ηsDqˉΞ(ηs)|].

    Proof. Using Lemma 3.2, property of modulus and the convexity property |ls=1ϖsl1s=1ξsDq| and |ls=1ϖsl1s=1ηsDq|, we have

    |Ω(ϖs;ηs;ξs)|=l1s=1(ηsξs)4[10|(1qϑ)||ls=1ϖsl1s=1ξsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ξs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))|dqϑ+10(1qϑ)|ls=1ϖsl1s=1ηsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ηs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))|dqϑ]l1s=1(ηsξs)4[10(1qϑ)(ls=1|ls=1ϖsl1s=1ξsDqˉΞ(ϖs)|2ϑ2l1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ξs)|ϑ2l1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ηs)|)dqϑ]+l1s=1(ηsξs)4[10(1qϑ)(ls=1|ls=1ϖsl1s=1ηsDqˉΞ(ϖs)|ϑ2l1s=1|ls=1ϖsl1s=1ηsDqˉΞ(ξs)|2ϑ2l1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ηs)|)dqϑ].

    After some calculations, we obtain our required result.

    If we choose l=2 in Theorem 3.5, then

    |ˉΞ(ϖ1+ϖ2ξ1)+ˉΞ(ϖ1+ϖ2η1)21η1ξ1[ϖ1+ϖ2ξ1+η12ϖ1+ϖ2η1ˉΞ(u)ϖ1+ϖ2η1dqu+ϖ1+ϖ2ξ1ϖ1+ϖ2ξ1+η12ˉΞ(u)ϖ1+ϖ2ξ1dqu]|η1ξ14[1[2]q(|ϖ1+ϖ2ξ1DqˉΞ(ϖ1)|+|ϖ1+ϖ2ξ1DqˉΞ(ϖ2)|)[2[3]q1]2[2]q[3]q|ϖ1+ϖ2ξ1DqˉΞ(ξ1)|12[2]q[3]q|ϖ1+ϖ2ξ1DqˉΞ(η1)|+1[2]q(|ϖ1+ϖ2η1DqˉΞ(ϖ1)|+|ϖ1+ϖ2η1DqˉΞ(ϖ2)|)12[2]q[3]q|ϖ1+ϖ2η1DqˉΞ(ξ1)|[2[3]q1]2[2]q[3]q|ϖ1+ϖ2η1DqˉΞ(η1)|],

    where 1r+1s=1.

    Theorem 3.6. Under the assumptions of Lemma 3.2 and the convexity property of |ls=1ϖsl1s=1ξsDq|r and |ls=1ϖsl1s=1ηsDq|r, we have

    |Ω(ϖs;ηs;ξs)|l1s=1(ηsξs)4(1[2]q)11r[(1[2]qls=1|ls=1ϖsl1s=1ξsDqˉΞ(ϖs)|r[2[3]q1]2[2]q[3]ql1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ξs)|r12[2]q[3]ql1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ηs)|r)1r+(1[2]qls=1|ls=1ϖsl1s=1ηsDqˉΞ(ϖs)|r12[2]q[3]ql1s=1|ls=1ϖsl1s=1ηsDqˉΞ(ξs)|r[2[3]q1]2[2]q[3]ql1s=1|ls=1ϖsl1s=1ηsDqˉΞ(ηs)|r)1r].

    Proof. Using Lemma 3.2, property of modulus, power-mean inequality and the convexity property of |ls=1ϖsl1s=1ξsDq|r and |ls=1ϖsl1s=1ηsDq|r, we have

    |Ω(ϖs;ηs;ξs)|l1s=1(ηsξs)4(10(1qϑ)dqϑ)11r×[(10(1qϑ)|ls=1ϖsl1s=1ξsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ξs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))|rdqϑ)1r+(10(1qϑ)|ls=1ϖsl1s=1ηsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ηs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))|rdqϑ)1r]l1s=1(ηsξs)4(q[2]q)1r[(10(1qϑ)(ls=1|ls=1ϖsl1s=1ξsDqˉΞ(ϖs)|r2ϑ2l1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ξs)|rϑ2l1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ηs)|r)dqϑ)1r+(10(1qϑ)(ls=1|ls=1ϖsl1s=1ηsDqˉΞ(ϖs)|rϑ2l1s=1|ls=1ϖsl1s=1ηsDqˉΞ(ξs)|r2ϑ2l1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ηs)|r)dqϑ)1r].

    After simple calculations, we achieve our final result.

    If we choose l=2 in Theorem 3.6, then

    |ˉΞ(ϖ1+ϖ2ξ1)+ˉΞ(ϖ1+ϖ2η1)21η1ξ1[ϖ1+ϖ2ξ1+η12ϖ1+ϖ2η1ˉΞ(u)ϖ1+ϖ2η1dqu+ϖ1+ϖ2ξ1ϖ1+ϖ2ξ1+η12ˉΞ(u)ϖ1+ϖ2ξ1dqu]|η1ξ14(1[2]q)11r[(1[2]q(|ϖ1+ϖ2ξ1DqˉΞ(ϖ1)|r+|ϖ1+ϖ2ξ1DqˉΞ(ϖ2)|r)[2[3]q1]2[2]q[3]q|ϖ1+ϖ2ξ1DqˉΞ(ξ1)|r12[2]q[3]q|ϖ1+ϖ2ξ1DqˉΞ(η1)|r)1r+(1[2]q(|ϖ1+ϖ2η1DqˉΞ(ϖ1)|r+|ϖ1+ϖ2η1DqˉΞ(ϖ2)|r)12[2]q[3]q|ϖ1+ϖ2η1DqˉΞ(ξ1)|r[2[3]q1]2[2]q[3]q|ϖ1+ϖ2η1DqˉΞ(η1)|r)1r],

    where 1r+1s=1.

    Theorem 3.7. Under the assumptions of Lemma 3.2 and if |ls=1ϖsl1s=1ξsDq|r and |ls=1ϖsl1s=1ηsDq|r are convex mappings, we have

    |Ω(ϖs;ηs;ξs)|l1s=1(ηsξs)4(10(1qϑ)rdqϑ)11r[(ls=1|ls=1ϖsl1s=1ξsDqˉΞ(ϖs)|r1+2q2[2]ql1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ξs)|r12[2]ql1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ηs)|r)1r+(ls=1|ls=1ϖsl1s=1ηsDqˉΞ(ϖs)|r12[2]ql1s=1|ls=1ϖsl1s=1ηsDqˉΞ(ξs)|r1+2q2[2]ql1s=1|ls=1ϖsl1s=1ηsDqˉΞ(ηs)|r)1r],

    where 1r+1s=1.

    Proof. Using Lemma 3.2, property of modulus, Hölder's inequality and using the convexity property of |ls=1ϖsl1s=1ξsDq|r and |ls=1ϖsl1s=1ηsDq|r, we have

    |ϑ(ϖs;ηs;ξs)|l1s=1(ηsξs)4(10(1qϑ)rdqϑ)11r×[(10|ls=1ϖsl1s=1ξsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ξs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))|rdqϑ)1r+(10|ls=1ϖsl1s=1ηsDqˉΞ((1ϑ)(ls=1ϖsl1s=1ηs)+ϑ(ls=1ϖsl1s=1ξs+ηs2))|rdqϑ)1r]l1s=1(ηsξs)4(10(1qϑ)rdqϑ)11r[(10(ls=1|ls=1ϖsl1s=1ξsDqˉΞ(ϖs)|r2ϑ2l1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ξs)|rϑ2l1s=1l1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ηs)|r)dqϑ)1r+(10(ls=1p|ls=1ϖsl1s=1ηsDqˉΞ(ϖs)|rϑ2l1s=1|ls=1ϖsl1s=1ηsDqˉΞ(ξs)|r2ϑ2l1s=1|ls=1ϖsl1s=1ξsDqˉΞ(ηs)|r)dqϑ)1r].

    After simple calculations, we achieve our final result.

    If we choose l=2 in Theorem 3.7, then

    |ˉΞ(ϖ1+ϖ2ξ1)+ˉΞ(ϖ1+ϖ2η1)21η1ξ1[ϖ1+ϖ2ξ1+η12ϖ1+ϖ2η1ˉΞ(u)ϖ1+ϖ2η1dqu+ϖ1+ϖ2ξ1ϖ1+ϖ2ξ1+η12ˉΞ(u)ϖ1+ϖ2ξ1dqu]|η1ξ14(10(1qϑ)rdqϑ)11r[(|ϖ1+ϖ2ξ1DqˉΞ(ϖ1)|r+|ϖ1+ϖ2ξ1DqˉΞ(ϖ2)|r1+2q2[2]q|ϖ1+ϖ2ξ1DqˉΞ(ξ1)|r12[2]q|ϖ1+ϖ2ξ1DqˉΞ(η1)|r)1r+(|ϖ1+ϖ2η1DqˉΞ(ϖ1)|r+|ϖ1+ϖ2η1DqˉΞ(ϖ2)|r12[2]q|ϖ1+ϖ2η1DqˉΞ(ξ1)|r1+2q2[2]q|ϖ1+ϖ2η1DqˉΞ(η1)|r)1r].

    Finally, we present some applications to bivariate means of non-negative real numbers in the support of main results. For more visualization, we check the validity through numeric examples and a graphical explanation is also mentioned. The arithmetic mean: A(ς1,ς2)=ς1+ς22,

    The generalized log-mean: Lp(ς1,ς2)=[ς2p+1ς1p+1(p+1)(ς2ς1)]1p.

    where pR{1,0}, ς1,ς2R, ς1ς2.

    Proposition 4.1. Assume that all the assumptions of Theorem 3.2 are held. Then

    |3[3]qL22(ϖ1+ϖ2η1,ϖ1+ϖ2ξ1+η12)+12[(ϖ1+ϖ2ξ1)2+(η1ξ1)24[3]qη1ξ1[2]q](ϖ1+ϖ2A(ξ1,η1))2|η1ξ14[q[2]q(B(ϖ1,ξ1)+B(ϖ2,ξ1))q[[3]q+q2]2[2]q[3]qB(ξ1,ξ1)q2[3]qB(η1,ξ1)+q[2]q(C(ϖ1,η1)+C(ϖ2,η1))q2[3]qC(ξ1,η1)q[[3]q+q2]2[2]q[3]qC(η1,η1)],

    where

    B(z,ξ1)=|(1+q)z+(1q)(ϖ1+ϖ2ξ1)|, (4.1)
    \begin{align} &C(z, \eta_{1}) = |(1+{\mathfrak{q}})z+(1-{\mathfrak{q}})(\varpi_{1}+\varpi_{2}-\eta_{1})|. \end{align} (4.2)

    Proof. The assertion follows directly from Theorem 3.2 for {{\bar{\Xi}}}(\mathfrak{u}) = \mathfrak{u}^2 .

    Now we give the numerical verification of Theorem 3.2.

    Example 4.1. Taking \varpi_{1} = 0, \xi_{1} = 1, \eta_{1} = 2 and \varpi_{2} = 3 in Proposition 4.1, we have 0.0595 < 0.5357 .

    For the graphical explanation of Theorem 3.2, we have

    \begin{align*} &\left|\frac{2+{\mathfrak{q}}+{\mathfrak{q}}^2}{4(1+{\mathfrak{q}}+{\mathfrak{q}}^2)}-\frac{1}{2(1+{\mathfrak{q}})}\right|\\ &\leq \frac{3{\mathfrak{q}}}{1+{\mathfrak{q}}}-\frac{3{\mathfrak{q}}(1+{\mathfrak{q}}+{\mathfrak{q}}^2)}{4(1+{\mathfrak{q}})(1+{\mathfrak{q}}+{\mathfrak{q}}^2)}-\frac{3{\mathfrak{q}}}{4(1+{\mathfrak{q}}+{\mathfrak{q}}^2)}. \end{align*}

    Figure 1 clearly emphasizes the correctness of Theorem 3.2, where the red and blue colours indicate the left-hand and right-hand sides respectively.

    Figure 1.   .

    Proposition 4.2. Assume that all the assumptions of Theorem 3.3 are held. Then

    \begin{align*} &\Bigg|\frac{3}{[3]_{{\mathfrak{q}}}}L_{2}^{2}\left(\varpi_{1}+\varpi_{2}-\eta_{1}, \varpi_{1}+\varpi_{2}-\frac{\xi_{1}+\eta_{1}}{2}\right)+\frac{1}{2}\left[(\varpi_{1}+\varpi_{2}-\xi_{1})^2+\frac{(\eta_{1}-\xi_{1})^2}{4[3]_{{\mathfrak{q}}}} -\frac{\eta_{1}-\xi_{1}}{[2]_{{\mathfrak{q}}}}\right]\\ &\quad-\left(\varpi_{1}+\varpi_{2}-A(\xi_{1}, \eta_{1})\right)^2\Bigg|\\ &\leq\frac{\eta_{1}-\xi_{1}}{4}\left(\frac{{\mathfrak{q}}}{[2]_{{\mathfrak{q}}}}\right)^{1-\frac{1}{r}}\left[\left(\frac{{\mathfrak{q}}}{[2]_{{\mathfrak{q}}}}\left(B^r(\varpi_{1}, \xi_{1})+B^r(\varpi_{2}, \xi_{1})\right) -\frac{{\mathfrak{q}}[[3]_{{\mathfrak{q}}}+{\mathfrak{q}}^2]}{2[2]_{{\mathfrak{q}}}[3]_{{\mathfrak{q}}}} B^r(\xi_{1}, \xi_{1})-\frac{{\mathfrak{q}}}{2[3]_{{\mathfrak{q}}}}B^r(\eta_{1}, \xi_{1})\right)^{\frac{1}{r}}\right.\nonumber\\ &\left.\quad+\left(\frac{{\mathfrak{q}}}{[2]_{{\mathfrak{q}}}}\left(C^r(\varpi_{1}, \eta_{1})+C^r(\varpi_{2}, \eta_{1})\right)- \frac{{\mathfrak{q}}}{2[3]_{{\mathfrak{q}}}}C^r(\xi_{1}, \eta_{1}) -\frac{{\mathfrak{q}}[[3]_{{\mathfrak{q}}}+{\mathfrak{q}}^2]}{2[2]_{{\mathfrak{q}}}[3]_{{\mathfrak{q}}}}C^r(\eta_{1}, \eta_{1})\right)^{\frac{1}{r}}\right], \end{align*}

    where

    \begin{align} &B^r(z, \xi_{1}) = |(1+{\mathfrak{q}})z+(1-{\mathfrak{q}})(\varpi_{1}+\varpi_{2}-\xi_{1})|^r. \end{align} (4.3)
    \begin{align} &C^r(z, \eta_{1}) = |(1+{\mathfrak{q}})z+(1-{\mathfrak{q}})(\varpi_{1}+\varpi_{2}-\eta_{1})|^r. \end{align} (4.4)

    Proof. The assertion follows directly from Theorem 3.3 for {{\bar{\Xi}}}(\mathfrak{u}) = \mathfrak{u}^2 .

    Now we give the numerical verification of Theorem 3.3.

    Example 4.2. Taking \varpi_{1} = 0, \xi_{1} = 1, \eta_{1} = 2 and \varpi_{2} = 3 in Proposition 4.2, we have 0.0595 < 0.7191 .

    For the graphical explanation of Theorem 3.3 are held. Then

    \begin{align*} &\left|\frac{2+{\mathfrak{q}}+{\mathfrak{q}}^2}{4(1+{\mathfrak{q}}+{\mathfrak{q}}^2)}-\frac{1}{2(1+{\mathfrak{q}})}\right|\\ &\leq \frac{1}{4}\left(\frac{{\mathfrak{q}}}{1+{\mathfrak{q}}}\right)\left[\left((2-2{\mathfrak{q}})^2+(5+{\mathfrak{q}})^2-\frac{(1+{\mathfrak{q}}+2{\mathfrak{q}}^2)(3-{\mathfrak{q}})^2}{2(1+{\mathfrak{q}}+{\mathfrak{q}}^2)}-\frac{8(1+{\mathfrak{q}})}{1+{\mathfrak{q}}+{\mathfrak{q}}^2}\right)^{\frac{1}{2}} \right.\\&\quad\left.+\left((1-{\mathfrak{q}})^2+(4+2{\mathfrak{q}})^2-\frac{2(1+{\mathfrak{q}})}{1+{\mathfrak{q}}+{\mathfrak{q}}^2}-\frac{(1+{\mathfrak{q}}+2{\mathfrak{q}}^2)(3+{\mathfrak{q}})^2}{2(1+{\mathfrak{q}}+{\mathfrak{q}}^2)}\right)^{\frac{1}{2}}\right]. \end{align*}

    Figure 2 clearly emphasizes the correctness of Theorem 3.3, where the red and blue colours indicate the left-hand and right-hand sides respectively.

    Figure 2.   .

    Proposition 4.3. Assume that all the assumptions of Theorem 3.4 are held. Then

    \begin{align*} &\Bigg|\frac{3}{[3]_{{\mathfrak{q}}}}L_{2}^{2}\left(\varpi_{1}+\varpi_{2}-\eta_{1}, \varpi_{1}+\varpi_{2}-\frac{\xi_{1}+\eta_{1}}{2}\right)+\frac{1}{2}\left[(\varpi_{1}+\varpi_{2}-\xi_{1})^2+\frac{(\eta_{1}-\xi_{1})^2}{4[3]_{{\mathfrak{q}}}} -\frac{\eta_{1}-\xi_{1}}{[2]_{{\mathfrak{q}}}}\right]\\ &\quad-\left(\varpi_{1}+\varpi_{2}-A(\xi_{1}, \eta_{1})\right)^2\Bigg|\\ &\leq \frac{\eta_{1}-\xi_{1}}{4}\left(\frac{{\mathfrak{q}}^r}{[r+1]_{{\mathfrak{q}}}}\right)^{1-\frac{1}{r}}\left[\left(B^r(\varpi_{1}, \xi_{1})+B^r(\varpi_{2}, \xi_{1})-\frac{1+2{\mathfrak{q}}}{2[2]_{{\mathfrak{q}}}} B^r(\xi_{1}, \xi_{1})-\frac{1}{2[2]_{{\mathfrak{q}}}}B^r(\eta_{1}, \xi_{1})\right)^{\frac{1}{r}}\right.\nonumber\\ &\left.\quad+\left(C^r(\varpi_{1}, \eta_{1})+C^r(\varpi_{2}, \eta_{1})- \frac{1}{2[2]_{{\mathfrak{q}}}}C^r(\xi_{1}, \eta_{1}) -\frac{1+2{\mathfrak{q}}}{2[2]_{{\mathfrak{q}}}}C^r(\eta_{1}, \eta_{1})\right)^{\frac{1}{r}}\right] \end{align*}

    B^r(z, \xi_{1}) and C^r(z, \eta_{1}) are defined by (4.3) and (4.4) respectively.

    Proof. The assertion follows directly from Theorem 3.4 for {{\bar{\Xi}}}(\mathfrak{u}) = \mathfrak{u}^2 .

    Now we give the numerical verification of Theorem 3.4.

    Example 4.3. Taking \varpi_{1} = 0, \xi_{1} = 1, \eta_{1} = 2 and \varpi_{2} = 3 in Proposition 4.3, we have 0.0595 < 0.8157 .

    For the graphical explanation of Theorem 3.4, we have following expression

    \begin{align*} &\left|\frac{2+{\mathfrak{q}}+{\mathfrak{q}}^2}{4(1+{\mathfrak{q}}+{\mathfrak{q}}^2)}-\frac{1}{2(1+{\mathfrak{q}})}\right|\\ &\leq \frac{1}{4}\left(\frac{{\mathfrak{q}}^2}{(1+{\mathfrak{q}}+{\mathfrak{q}}^2)}\right)^{\frac{1}{2}}\left[\left((2-2{\mathfrak{q}})^2+(5+{\mathfrak{q}})^2-\frac{(1+2{\mathfrak{q}})(3-{\mathfrak{q}})^2}{2(1+{\mathfrak{q}})}-\frac{8}{1+{\mathfrak{q}}}\right)^{\frac{1}{2}} \right.\\&\quad\left.+\left((1-{\mathfrak{q}})^2+(4+2{\mathfrak{q}})^2-\frac{2}{1+{\mathfrak{q}}}-\frac{(1+2{\mathfrak{q}})(3+{\mathfrak{q}})^2}{2(1+{\mathfrak{q}})}\right)^{\frac{1}{2}}\right]. \end{align*}

    Figure 3 clearly emphasizes the correctness of Theorem 3.4, where the red and blue colours indicate the left-hand and right-hand sides respectively.

    Figure 3.   .

    Proposition 4.4. Assume that all the assumptions of Theorem 3.5 are held. Then

    \begin{align*} &\left|A(\varpi_{1}+\varpi_{2}-\xi_{1}, \varpi_{1}+\varpi_{2}-\eta_{1})-\frac{3}{[3]_{{\mathfrak{q}}}}L_{2}^{2}\left(\varpi_{1}+\varpi_{2}-\eta_{1}, \varpi_{1}+\varpi_{2} -\frac{\xi_{1}+\eta_{1}}{2}\right)\right.\\ &\left.-\frac{1}{2}\left[(\varpi_{1}+\varpi_{2}-\xi_{1})^2+\frac{(\eta_{1}-\xi_{1})^2}{4[3]_{{\mathfrak{q}}}} -\frac{\eta_{1}-\xi_{1}}{[2]_{{\mathfrak{q}}}}\right]\right|\\ &\leq\frac{\eta_{1}-\xi_{1}}{4}\left[\frac{1}{[2]_{{\mathfrak{q}}}}\left(B(\varpi_{1}, \xi_{1})+B(\varpi_{2}, \xi_{1})\right)-\frac{[2[3]_{{\mathfrak{q}}}-1]}{2[2]_{{\mathfrak{q}}}[3]_{{\mathfrak{q}}}} B(\xi_{1}, \xi_{1})-\frac{1}{2[2]_{{\mathfrak{q}}}[3]_{{\mathfrak{q}}}}B(\eta_{1}, \xi_{1})\right.\nonumber\\ &\left.\quad+\frac{1}{[2]_{{\mathfrak{q}}}}\left(C(\varpi_{1}, \eta_{1})+C(\varpi_{2}, \eta_{1})\right)-\frac{1}{2[2]_{{\mathfrak{q}}}[3]_{{\mathfrak{q}}}} C(\xi_{1}, \eta_{1}) -\frac{[2[3]_{{\mathfrak{q}}}-1]}{2[2]_{{\mathfrak{q}}}[3]_{{\mathfrak{q}}}}C(\eta_{1}, \eta_{1})\right], \end{align*}

    where B(z, \xi_{1}) and C(z, \eta_{1}) are defined by (4.1) and (4.2) respectively.

    Proof. The assertion follows directly from Theorem 3.5 for {{\bar{\Xi}}}(\mathfrak{u}) = \mathfrak{u}^2 .

    Now we give the numerical verification of Theorem 3.5.

    Example 4.4. Taking \varpi_{1} = 0, \xi_{1} = 1, \eta_{1} = 2 and \varpi_{2} = 3 in Proposition 4.4, we have 0.1905 < 1.0000 .

    For the graphical explanation of Theorem 3.5, we have following expression

    \begin{align*} \left|\frac{1}{2(1+{\mathfrak{q}})}-\frac{1}{4(1+{\mathfrak{q}}+{\mathfrak{q}}^2)}\right| \leq \frac{3}{1+{\mathfrak{q}}}-\frac{3}{2(1+{\mathfrak{q}})}. \end{align*}

    Figure 4 clearly emphasizes the correctness of Theorem 3.5, where the red and blue colours indicate the left-hand and right-hand sides respectively.

    Figure 4.   .

    Proposition 4.5. Assume that all the assumptions of Theorem 3.6 are held, Then

    \begin{align*} &\left|A(\varpi_{1}+\varpi_{2}-\xi_{1}, \varpi_{1}+\varpi_{2}-\eta_{1})-\frac{3}{[3]_{{\mathfrak{q}}}}L_{2}^{2}\left(\varpi_{1}+\varpi_{2}-\eta_{1}, \varpi_{1}+\varpi_{2} -\frac{\xi_{1}+\eta_{1}}{2}\right)\right.\\ &\left.-\frac{1}{2}\left[(\varpi_{1}+\varpi_{2}-\xi_{1})^2+\frac{(\eta_{1}-\xi_{1})^2}{4[3]_{{\mathfrak{q}}}} -\frac{\eta_{1}-\xi_{1}}{[2]_{{\mathfrak{q}}}}\right]\right|\\ &\leq\frac{\eta_{1}-\xi_{1}}{4}\left(\frac{1}{[2]_{{\mathfrak{q}}}}\right)^{1-\frac{1}{r}}\left[\left(\frac{1}{[2]_{{\mathfrak{q}}}}\left(B^r(\varpi_{1}, \xi_{1})+B^r(\varpi_{2}, \xi_{1})\right) -\frac{[2[3]_{{\mathfrak{q}}}-1]}{2[2]_{{\mathfrak{q}}}[3]_{{\mathfrak{q}}}} B^r(\xi_{1}, \xi_{1})-\frac{1}{2[2]_{{\mathfrak{q}}}[3]_{{\mathfrak{q}}}}B^r(\eta_{1}, \xi_{1})\right)^{\frac{1}{r}}\right.\nonumber\\ &\left.\quad+\left(\frac{1}{[2]_{{\mathfrak{q}}}}\left(B^r(\varpi_{1}, \eta_{1})+B^r(\varpi_{2}, \eta_{1})\right)- \frac{1}{2[2]_{{\mathfrak{q}}}[3]_{{\mathfrak{q}}}}B^r(\xi_{1}, \eta_{1}) -\frac{[2[3]_{{\mathfrak{q}}}-1]}{2[2]_{{\mathfrak{q}}}[3]_{{\mathfrak{q}}}}B^r(\eta_{1}, \eta_{1})\right)^{\frac{1}{r}}\right], \end{align*}

    where B^r(z, \xi_{1}) and B^r(z, \eta_{1}) are defined by (4.3) and (4.4) respectively.

    Proof. The assertion follows directly from Theorem 3.6 for {{\bar{\Xi}}}(\mathfrak{u}) = \mathfrak{u}^2 .

    Now we give the numerical verification of Theorem 3.6.

    Example 4.5. Taking \varpi_{1} = 0, \xi_{1} = 1, \eta_{1} = 2 and \varpi_{2} = 3 in Proposition 4.5, we have 0.1905 < 1.4387 .

    For the graphical explanation of Theorem 3.6, we have following expression

    \begin{align*} &\left|\frac{1}{2(1+{\mathfrak{q}})}-\frac{1}{4(1+{\mathfrak{q}}+{\mathfrak{q}}^2)}\right|\\ &\leq \frac{1}{4}\left(\frac{1}{1+{\mathfrak{q}}}\right)\left[\left((2-2{\mathfrak{q}})^2+(5+{\mathfrak{q}})^2-\frac{(2(1+{\mathfrak{q}}+{\mathfrak{q}}^2)-1)(3-{\mathfrak{q}})^2}{2(1+{\mathfrak{q}}+{\mathfrak{q}}^2)}-\frac{8}{1+{\mathfrak{q}}+{\mathfrak{q}}^2}\right)^{\frac{1}{2}}\right.\\ &\left.+\left((1-{\mathfrak{q}})^2+(4+2{\mathfrak{q}})^2-\frac{2}{1+{\mathfrak{q}}+{\mathfrak{q}}^2}-\frac{(2(1+{\mathfrak{q}}+{\mathfrak{q}}^2)-1)(3+{\mathfrak{q}})^2}{2(1+{\mathfrak{q}}+{\mathfrak{q}}^2)}\right)^{\frac{1}{2}} \right]. \end{align*}

    Figure 5 clearly emphasizes the correctness of Theorem 3.6, where the red and blue colours indicate the left-hand and right-hand sides respectively.

    Figure 5.   .

    Proposition 4.6. Assume that all the assumptions of Theorem 3.7 are held. Then

    \begin{align*} &\left|A(\varpi_{1}+\varpi_{2}-\xi_{1}, \varpi_{1}+\varpi_{2}-\eta_{1})-\frac{3}{[3]_{{\mathfrak{q}}}}L_{2}^{2}\left(\varpi_{1}+\varpi_{2}-\eta_{1}, \varpi_{1} +\varpi_{2}-\frac{\xi_{1}+\eta_{1}}{2}\right)\right.\\ &\left.-\frac{1}{2}\left[(\varpi_{1}+\varpi_{2}-\xi_{1})^2+\frac{(\eta_{1}-\xi_{1})^2}{4[3]_{{\mathfrak{q}}}} -\frac{\eta_{1}-\xi_{1}}{[2]_{{\mathfrak{q}}}}\right]\right|\\ &\leq\frac{\eta_{1}-\xi_{1}}{4}\left(\int_0^1(1-\mathfrak{q}{\vartheta})^r\mathrm{d}_{{\mathfrak{q}}}{\vartheta}\right)^{1-\frac{1}{r}}\left[\left(B^r(\varpi_{1}, \xi_{1})+B^r(\varpi_{2}, \xi_{1})-\frac{1+2{\mathfrak{q}}}{2[2]_{{\mathfrak{q}}}} B^r(\xi_{1}, \xi_{1})-\frac{1}{2[2]_{{\mathfrak{q}}}}B^r(\eta_{1}, \xi_{1})\right)^{\frac{1}{r}}\right.\nonumber\\ &\left.\quad+\left(C^r(\varpi_{1}, \eta_{1})+C^r(\varpi_{2}, \eta_{1})- \frac{1}{2[2]_{{\mathfrak{q}}}}C^r(\xi_{1}, \eta_{1}) -\frac{1+2{\mathfrak{q}}}{2[2]_{{\mathfrak{q}}}}C^r(\eta_{1}, \eta_{1})\right)^{\frac{1}{r}}\right], \end{align*}

    B^r(z, \xi_{1}) and B^r(z, \eta_{1}) are defined by (4.3) and (4.4) respectively.

    Proof. The assertion follows directly from Theorem 3.7 for {{\bar{\Xi}}}(\mathfrak{u}) = \mathfrak{u}^2 .

    Now we give the numerical verification of Theorem 3.7.

    Example 4.6. Taking \varpi_{1} = 0, \xi_{1} = 1, \eta_{1} = 2 and \varpi_{2} = 3 in Proposition 4.6, we have 0.1905 < 1.4892 .

    For the graphical explanation of Theorem 3.7, we have following expression

    \begin{align*} &\left|\frac{1}{2(1+{\mathfrak{q}})}-\frac{1}{4(1+{\mathfrak{q}}+{\mathfrak{q}}^2)}\right|\\ &\leq \frac{1}{4}\left(1+\frac{{\mathfrak{q}}^2}{(1+{\mathfrak{q}}+{\mathfrak{q}}^2)}-\frac{2{\mathfrak{q}}}{1+{\mathfrak{q}}}\right)^{\frac{1}{2}}\left[\left((2-2{\mathfrak{q}})^2+(5+{\mathfrak{q}})^2-\frac{(1+2{\mathfrak{q}})(3-{\mathfrak{q}})^2}{2(1+{\mathfrak{q}})}-\frac{8}{1+{\mathfrak{q}}}\right)^{\frac{1}{2}} \right.\\&\quad\left.+\left((1-{\mathfrak{q}})^2+(4+2{\mathfrak{q}})^2-\frac{2}{1+{\mathfrak{q}}}-\frac{(1+2{\mathfrak{q}})(3+{\mathfrak{q}})^2}{2(1+{\mathfrak{q}})}\right)^{\frac{1}{2}}\right]. \end{align*}

    Figure 6 clearly emphasizes the correctness of Theorem 3.7, where the red and blue colours indicate the left-hand and right-hand sides respectively.

    Figure 6.   .

    The Trapezium-Jensen-Mercer (TJM) inequality is a well-researched and extensively studied result in the literature. Various versions of this inequality have been derived using different concepts of convexity, including fractional and {\mathfrak{q}} -calculus. In this study, we have introduced new continuous and discrete quantum versions of TJM and established some new bounds of inequality through convex mapping. Additionally, we have provided several applications and graphical analyses to support our findings. Moving forward, we plan to derive {\mathfrak{q}} -fractional and (p, {\mathfrak{q}}) -analogues of TJM, Simpson-Mercer, and Ostrowski-like inequalities that involve different categories of convexity.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors are thankful to the editor and anonymous reviewers for their valuable comments and suggestions. This research is supported by "Researchers Supporting Project number (RSP2023R158), King Saud University, Riyadh, Saudi Arabia." Muhammad Uzair Awan is thankful to HEC Pakistan for 8081/Punjab/NRPU/R & D/HEC/2017.

    The authors declare that they have no competing interests.



    [1] P. Schober, T. R. Vetter, Chi-square tests in medical research, Anesth. Analg., 129 (2019), 1193. https://doi.org/10.1213/ANE.0000000000004410 doi: 10.1213/ANE.0000000000004410
    [2] M. V. Koutras, S. Bersimis, D. L. Antzoulakos, Improving the performance of the chi-square control chart via runs rules, Methodol. Comput. Appl. Probab., 8 (2006), 409–426. https://doi.org/10.1007/s11009-006-9754-z doi: 10.1007/s11009-006-9754-z
    [3] N. T. Thomopoulos, Essentials of Monte Carlo simulation: Statistical methods for building simulation models, New York: Springer, 2013. https://doi.org/10.1007/978-1-4614-6022-0
    [4] M. A. Jdid, R. Alhabib, A. A. Salama, Fundamentals of neutrosophical simulation for generating random numbers associated with uniform probability distribution, Neutrosophic Sets Sy., 49 (2022), 92–102. https://doi.org/10.5281/zenodo.6426375 doi: 10.5281/zenodo.6426375
    [5] M. A. Jdid, R. Alhabib, A. A. Salama, The static model of inventory management without a deficit with Neutrosophic logic, International Journal of Neutrosophic Science, 16 (2021), 42–48. https://doi.org/10.54216/IJNS.160104 doi: 10.54216/IJNS.160104
    [6] J. F. Monahan, An algorithm for generating chi random variables, ACM T. Math. Software, 13 (1987), 168–172. https://doi.org/10.1145/328512.328522 doi: 10.1145/328512.328522
    [7] E. Shmerling, Algorithms for generating random variables with a rational probability-generating function, Int. J. Comput. Math., 92 (2015), 2001–2010. https://doi.org/10.1080/00207160.2014.945918 doi: 10.1080/00207160.2014.945918
    [8] N. Ortigosa, M. Orellana-Panchame, J. C. Castro-Palacio, P. F. de Córdoba, J. M. Isidro, Monte Carlo simulation of a modified Chi distribution considering asymmetry in the generating functions: Application to the study of health-related variables, Symmetry, 13 (2021), 924. https://doi.org/10.3390/sym13060924 doi: 10.3390/sym13060924
    [9] L. Devroye, A simple algorithm for generating random variates with a log-concave density, Computing, 33 (1984), 247–257. https://doi.org/10.1007/BF02242271 doi: 10.1007/BF02242271
    [10] L. Devroye, Nonuniform random variate generation, Handbooks in Operations Research and Management Science, 13 (2006), 83–121. https://doi.org/10.1016/S0927-0507(06)13004-2 doi: 10.1016/S0927-0507(06)13004-2
    [11] L. Devroye, Random variate generation for the generalized inverse Gaussian distribution, Stat. Comput., 24 (2014), 239–246. https://doi.org/10.1007/s11222-012-9367-z doi: 10.1007/s11222-012-9367-z
    [12] E. A. Luengo, Gamma Pseudo random number generators, ACM Comput. Surv., 55 (2022), 1–33. https://doi.org/10.1145/3527157 doi: 10.1145/3527157
    [13] H. Yao, T. Taimre, Estimating tail probabilities of random sums of phase-type scale mixture random variables, Algorithms, 15 (2022), 350. https://doi.org/10.3390/a15100350 doi: 10.3390/a15100350
    [14] T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to algorithms, Massachusetts: MIT Press, 2022.
    [15] D. H. Pereira, Itamaracá: A novel simple way to generate Pseudo-random numbers, Cambridge Open Engage, 2022. https://doi.org/10.33774/coe-2022-zsw6t doi: 10.33774/coe-2022-zsw6t
    [16] F. Smarandache, Introduction to neutrosophic statistics, Craiova: Romania-Educational Publisher, 2014.
    [17] F. Smarandache, Neutrosophic statistics is an extension of interval statistics, while Plithogenic Statistics is the most general form of statistics (second version), International Journal of Neutrosophic Science, 19 (2022), 148–165. https://doi.org/10.54216/ IJNS.190111 doi: 10.54216/IJNS.190111
    [18] J. Q. Chen, J. Ye, S. G. Du, Scale effect and anisotropy analyzed for neutrosophic numbers of rock joint roughness coefficient based on neutrosophic statistics, Symmetry, 9 (2017), 208. https://doi.org/10.3390/sym9100208 doi: 10.3390/sym9100208
    [19] J. Chen, J. Ye, S. G. Du, R. Yong, Expressions of rock joint roughness coefficient using neutrosophic interval statistical numbers, Symmetry, 9 (2017), 123. https://doi.org/10.3390/sym9070123 doi: 10.3390/sym9070123
    [20] M. Aslam, Truncated variable algorithm using DUS-neutrosophic Weibull distribution, Complex Intell. Syst., 9 (2023), 3107–3114.
    [21] F. Smarandache, Neutrosophic statistics is an extension of interval statistics, while Plithogenic statistics is the most general form of statistics, Neutrosophic Computing and Machine Learning, 23 (2022), 21–38.
    [22] R. Alhabib, M. M. Ranna, H. Farah, A. A. Salama, Some neutrosophic probability distributions, Neutrosophic Sets Sy., 22 (2018), 30–38.
    [23] Z. Khan, A. Al-Bossly, M. M. A. Almazah, F. S. Alduais, On statistical development of neutrosophic gamma distribution with applications to complex data analysis, Complexity, 2021 (2021), 3701236. https://doi.org/10.1155/2021/3701236 doi: 10.1155/2021/3701236
    [24] R. A. K. Sherwani, M. Aslam, M. A. Raza, M. Farooq, M. Abid, M. Tahir, Neutrosophic normal probability distribution–A spine of parametric neutrosophic statistical tests: properties and applications, In: Neutrosophic operational research, Cham: Springer, 2021,153–169. https://doi.org/10.1007/978-3-030-57197-9_8
    [25] C. Granados, Some discrete neutrosophic distributions with neutrosophic parameters based on neutrosophic random variables, Hacet. J. Math. Stat., 51 (2022), 1442–1457. https://doi.org/10.15672/hujms.1099081 doi: 10.15672/hujms.1099081
    [26] C. Granados, A. K. Das, B. Das, Some continuous neutrosophic distributions with neutrosophic parameters based on neutrosophic random variables, Advances in the Theory of Nonlinear Analysis and its Application, 6 (2023), 380–389.
    [27] Y. H. Guo, A. Sengur, NCM: Neutrosophic c-means clustering algorithm, Pattern Recogn., 48 (2015), 2710–2724. https://doi.org/10.1016/j.patcog.2015.02.018 doi: 10.1016/j.patcog.2015.02.018
    [28] H. Garg, Nancy, Algorithms for single-valued neutrosophic decision making based on TOPSIS and clustering methods with new distance measure, AIMS Mathematics, 5 (2020), 2671–2693. https://doi.org/10.3934/math.2020173 doi: 10.3934/math.2020173
    [29] M. Aslam, Simulating imprecise data: sine-cosine and convolution methods with neutrosophic normal distribution, J. Big Data, 10 (2023), 143. https://doi.org/10.1186/s40537-023-00822-4 doi: 10.1186/s40537-023-00822-4
    [30] M. Aslam, Uncertainty-driven generation of neutrosophic random variates from the Weibull distribution, J. Big Data, 10 (2023), 177. https://doi.org/10.1186/s40537-023-00860-y doi: 10.1186/s40537-023-00860-y
    [31] M. Aslam, F. S. Alamri, Algorithm for generating neutrosophic data using accept-reject method, J. Big Data, 10 (2023), 175. https://doi.org/10.1186/s40537-023-00855-9 doi: 10.1186/s40537-023-00855-9
    [32] M. Catelani, A. Zanobini, L. Ciani, Uncertainty interval evaluation using the Chi-square and Fisher distributions in the measurement process, Metrol. Meas. Syst., 17 (2010), 195–204. https://doi.org/10.2478/v10178-010-0017-5 doi: 10.2478/v10178-010-0017-5
    [33] R. H. Hariri, E. M. Fredericks, K. M. Bowers, Uncertainty in big data analytics: survey, opportunities, and challenges, J. Big Data, 6 (2019), 44. https://doi.org/10.1186/s40537-019-0206-3 doi: 10.1186/s40537-019-0206-3
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